Identifier
-
Mp00017:
Binary trees
—to 312-avoiding permutation⟶
Permutations
St000054: Permutations ⟶ ℤ
Values
=>
Cc0010;cc-rep-0
[.,.]=>[1]=>1
[.,[.,.]]=>[2,1]=>2
[[.,.],.]=>[1,2]=>1
[.,[.,[.,.]]]=>[3,2,1]=>3
[.,[[.,.],.]]=>[2,3,1]=>2
[[.,.],[.,.]]=>[1,3,2]=>1
[[.,[.,.]],.]=>[2,1,3]=>2
[[[.,.],.],.]=>[1,2,3]=>1
[.,[.,[.,[.,.]]]]=>[4,3,2,1]=>4
[.,[.,[[.,.],.]]]=>[3,4,2,1]=>3
[.,[[.,.],[.,.]]]=>[2,4,3,1]=>2
[.,[[.,[.,.]],.]]=>[3,2,4,1]=>3
[.,[[[.,.],.],.]]=>[2,3,4,1]=>2
[[.,.],[.,[.,.]]]=>[1,4,3,2]=>1
[[.,.],[[.,.],.]]=>[1,3,4,2]=>1
[[.,[.,.]],[.,.]]=>[2,1,4,3]=>2
[[[.,.],.],[.,.]]=>[1,2,4,3]=>1
[[.,[.,[.,.]]],.]=>[3,2,1,4]=>3
[[.,[[.,.],.]],.]=>[2,3,1,4]=>2
[[[.,.],[.,.]],.]=>[1,3,2,4]=>1
[[[.,[.,.]],.],.]=>[2,1,3,4]=>2
[[[[.,.],.],.],.]=>[1,2,3,4]=>1
[.,[.,[.,[.,[.,.]]]]]=>[5,4,3,2,1]=>5
[.,[.,[.,[[.,.],.]]]]=>[4,5,3,2,1]=>4
[.,[.,[[.,.],[.,.]]]]=>[3,5,4,2,1]=>3
[.,[.,[[.,[.,.]],.]]]=>[4,3,5,2,1]=>4
[.,[.,[[[.,.],.],.]]]=>[3,4,5,2,1]=>3
[.,[[.,.],[.,[.,.]]]]=>[2,5,4,3,1]=>2
[.,[[.,.],[[.,.],.]]]=>[2,4,5,3,1]=>2
[.,[[.,[.,.]],[.,.]]]=>[3,2,5,4,1]=>3
[.,[[[.,.],.],[.,.]]]=>[2,3,5,4,1]=>2
[.,[[.,[.,[.,.]]],.]]=>[4,3,2,5,1]=>4
[.,[[.,[[.,.],.]],.]]=>[3,4,2,5,1]=>3
[.,[[[.,.],[.,.]],.]]=>[2,4,3,5,1]=>2
[.,[[[.,[.,.]],.],.]]=>[3,2,4,5,1]=>3
[.,[[[[.,.],.],.],.]]=>[2,3,4,5,1]=>2
[[.,.],[.,[.,[.,.]]]]=>[1,5,4,3,2]=>1
[[.,.],[.,[[.,.],.]]]=>[1,4,5,3,2]=>1
[[.,.],[[.,.],[.,.]]]=>[1,3,5,4,2]=>1
[[.,.],[[.,[.,.]],.]]=>[1,4,3,5,2]=>1
[[.,.],[[[.,.],.],.]]=>[1,3,4,5,2]=>1
[[.,[.,.]],[.,[.,.]]]=>[2,1,5,4,3]=>2
[[.,[.,.]],[[.,.],.]]=>[2,1,4,5,3]=>2
[[[.,.],.],[.,[.,.]]]=>[1,2,5,4,3]=>1
[[[.,.],.],[[.,.],.]]=>[1,2,4,5,3]=>1
[[.,[.,[.,.]]],[.,.]]=>[3,2,1,5,4]=>3
[[.,[[.,.],.]],[.,.]]=>[2,3,1,5,4]=>2
[[[.,.],[.,.]],[.,.]]=>[1,3,2,5,4]=>1
[[[.,[.,.]],.],[.,.]]=>[2,1,3,5,4]=>2
[[[[.,.],.],.],[.,.]]=>[1,2,3,5,4]=>1
[[.,[.,[.,[.,.]]]],.]=>[4,3,2,1,5]=>4
[[.,[.,[[.,.],.]]],.]=>[3,4,2,1,5]=>3
[[.,[[.,.],[.,.]]],.]=>[2,4,3,1,5]=>2
[[.,[[.,[.,.]],.]],.]=>[3,2,4,1,5]=>3
[[.,[[[.,.],.],.]],.]=>[2,3,4,1,5]=>2
[[[.,.],[.,[.,.]]],.]=>[1,4,3,2,5]=>1
[[[.,.],[[.,.],.]],.]=>[1,3,4,2,5]=>1
[[[.,[.,.]],[.,.]],.]=>[2,1,4,3,5]=>2
[[[[.,.],.],[.,.]],.]=>[1,2,4,3,5]=>1
[[[.,[.,[.,.]]],.],.]=>[3,2,1,4,5]=>3
[[[.,[[.,.],.]],.],.]=>[2,3,1,4,5]=>2
[[[[.,.],[.,.]],.],.]=>[1,3,2,4,5]=>1
[[[[.,[.,.]],.],.],.]=>[2,1,3,4,5]=>2
[[[[[.,.],.],.],.],.]=>[1,2,3,4,5]=>1
[.,[.,[.,[.,[.,[.,.]]]]]]=>[6,5,4,3,2,1]=>6
[.,[.,[.,[.,[[.,.],.]]]]]=>[5,6,4,3,2,1]=>5
[.,[.,[.,[[.,.],[.,.]]]]]=>[4,6,5,3,2,1]=>4
[.,[.,[.,[[.,[.,.]],.]]]]=>[5,4,6,3,2,1]=>5
[.,[.,[.,[[[.,.],.],.]]]]=>[4,5,6,3,2,1]=>4
[.,[.,[[.,.],[.,[.,.]]]]]=>[3,6,5,4,2,1]=>3
[.,[.,[[.,.],[[.,.],.]]]]=>[3,5,6,4,2,1]=>3
[.,[.,[[.,[.,.]],[.,.]]]]=>[4,3,6,5,2,1]=>4
[.,[.,[[[.,.],.],[.,.]]]]=>[3,4,6,5,2,1]=>3
[.,[.,[[.,[.,[.,.]]],.]]]=>[5,4,3,6,2,1]=>5
[.,[.,[[.,[[.,.],.]],.]]]=>[4,5,3,6,2,1]=>4
[.,[.,[[[.,.],[.,.]],.]]]=>[3,5,4,6,2,1]=>3
[.,[.,[[[.,[.,.]],.],.]]]=>[4,3,5,6,2,1]=>4
[.,[.,[[[[.,.],.],.],.]]]=>[3,4,5,6,2,1]=>3
[.,[[.,.],[.,[.,[.,.]]]]]=>[2,6,5,4,3,1]=>2
[.,[[.,.],[.,[[.,.],.]]]]=>[2,5,6,4,3,1]=>2
[.,[[.,.],[[.,.],[.,.]]]]=>[2,4,6,5,3,1]=>2
[.,[[.,.],[[.,[.,.]],.]]]=>[2,5,4,6,3,1]=>2
[.,[[.,.],[[[.,.],.],.]]]=>[2,4,5,6,3,1]=>2
[.,[[.,[.,.]],[.,[.,.]]]]=>[3,2,6,5,4,1]=>3
[.,[[.,[.,.]],[[.,.],.]]]=>[3,2,5,6,4,1]=>3
[.,[[[.,.],.],[.,[.,.]]]]=>[2,3,6,5,4,1]=>2
[.,[[[.,.],.],[[.,.],.]]]=>[2,3,5,6,4,1]=>2
[.,[[.,[.,[.,.]]],[.,.]]]=>[4,3,2,6,5,1]=>4
[.,[[.,[[.,.],.]],[.,.]]]=>[3,4,2,6,5,1]=>3
[.,[[[.,.],[.,.]],[.,.]]]=>[2,4,3,6,5,1]=>2
[.,[[[.,[.,.]],.],[.,.]]]=>[3,2,4,6,5,1]=>3
[.,[[[[.,.],.],.],[.,.]]]=>[2,3,4,6,5,1]=>2
[.,[[.,[.,[.,[.,.]]]],.]]=>[5,4,3,2,6,1]=>5
[.,[[.,[.,[[.,.],.]]],.]]=>[4,5,3,2,6,1]=>4
[.,[[.,[[.,.],[.,.]]],.]]=>[3,5,4,2,6,1]=>3
[.,[[.,[[.,[.,.]],.]],.]]=>[4,3,5,2,6,1]=>4
[.,[[.,[[[.,.],.],.]],.]]=>[3,4,5,2,6,1]=>3
[.,[[[.,.],[.,[.,.]]],.]]=>[2,5,4,3,6,1]=>2
[.,[[[.,.],[[.,.],.]],.]]=>[2,4,5,3,6,1]=>2
[.,[[[.,[.,.]],[.,.]],.]]=>[3,2,5,4,6,1]=>3
[.,[[[[.,.],.],[.,.]],.]]=>[2,3,5,4,6,1]=>2
[.,[[[.,[.,[.,.]]],.],.]]=>[4,3,2,5,6,1]=>4
[.,[[[.,[[.,.],.]],.],.]]=>[3,4,2,5,6,1]=>3
[.,[[[[.,.],[.,.]],.],.]]=>[2,4,3,5,6,1]=>2
[.,[[[[.,[.,.]],.],.],.]]=>[3,2,4,5,6,1]=>3
[.,[[[[[.,.],.],.],.],.]]=>[2,3,4,5,6,1]=>2
[[.,.],[.,[.,[.,[.,.]]]]]=>[1,6,5,4,3,2]=>1
[[.,.],[.,[.,[[.,.],.]]]]=>[1,5,6,4,3,2]=>1
[[.,.],[.,[[.,.],[.,.]]]]=>[1,4,6,5,3,2]=>1
[[.,.],[.,[[.,[.,.]],.]]]=>[1,5,4,6,3,2]=>1
[[.,.],[.,[[[.,.],.],.]]]=>[1,4,5,6,3,2]=>1
[[.,.],[[.,.],[.,[.,.]]]]=>[1,3,6,5,4,2]=>1
[[.,.],[[.,.],[[.,.],.]]]=>[1,3,5,6,4,2]=>1
[[.,.],[[.,[.,.]],[.,.]]]=>[1,4,3,6,5,2]=>1
[[.,.],[[[.,.],.],[.,.]]]=>[1,3,4,6,5,2]=>1
[[.,.],[[.,[.,[.,.]]],.]]=>[1,5,4,3,6,2]=>1
[[.,.],[[.,[[.,.],.]],.]]=>[1,4,5,3,6,2]=>1
[[.,.],[[[.,.],[.,.]],.]]=>[1,3,5,4,6,2]=>1
[[.,.],[[[.,[.,.]],.],.]]=>[1,4,3,5,6,2]=>1
[[.,.],[[[[.,.],.],.],.]]=>[1,3,4,5,6,2]=>1
[[.,[.,.]],[.,[.,[.,.]]]]=>[2,1,6,5,4,3]=>2
[[.,[.,.]],[.,[[.,.],.]]]=>[2,1,5,6,4,3]=>2
[[.,[.,.]],[[.,.],[.,.]]]=>[2,1,4,6,5,3]=>2
[[.,[.,.]],[[.,[.,.]],.]]=>[2,1,5,4,6,3]=>2
[[.,[.,.]],[[[.,.],.],.]]=>[2,1,4,5,6,3]=>2
[[[.,.],.],[.,[.,[.,.]]]]=>[1,2,6,5,4,3]=>1
[[[.,.],.],[.,[[.,.],.]]]=>[1,2,5,6,4,3]=>1
[[[.,.],.],[[.,.],[.,.]]]=>[1,2,4,6,5,3]=>1
[[[.,.],.],[[.,[.,.]],.]]=>[1,2,5,4,6,3]=>1
[[[.,.],.],[[[.,.],.],.]]=>[1,2,4,5,6,3]=>1
[[.,[.,[.,.]]],[.,[.,.]]]=>[3,2,1,6,5,4]=>3
[[.,[.,[.,.]]],[[.,.],.]]=>[3,2,1,5,6,4]=>3
[[.,[[.,.],.]],[.,[.,.]]]=>[2,3,1,6,5,4]=>2
[[.,[[.,.],.]],[[.,.],.]]=>[2,3,1,5,6,4]=>2
[[[.,.],[.,.]],[.,[.,.]]]=>[1,3,2,6,5,4]=>1
[[[.,.],[.,.]],[[.,.],.]]=>[1,3,2,5,6,4]=>1
[[[.,[.,.]],.],[.,[.,.]]]=>[2,1,3,6,5,4]=>2
[[[.,[.,.]],.],[[.,.],.]]=>[2,1,3,5,6,4]=>2
[[[[.,.],.],.],[.,[.,.]]]=>[1,2,3,6,5,4]=>1
[[[[.,.],.],.],[[.,.],.]]=>[1,2,3,5,6,4]=>1
[[.,[.,[.,[.,.]]]],[.,.]]=>[4,3,2,1,6,5]=>4
[[.,[.,[[.,.],.]]],[.,.]]=>[3,4,2,1,6,5]=>3
[[.,[[.,.],[.,.]]],[.,.]]=>[2,4,3,1,6,5]=>2
[[.,[[.,[.,.]],.]],[.,.]]=>[3,2,4,1,6,5]=>3
[[.,[[[.,.],.],.]],[.,.]]=>[2,3,4,1,6,5]=>2
[[[.,.],[.,[.,.]]],[.,.]]=>[1,4,3,2,6,5]=>1
[[[.,.],[[.,.],.]],[.,.]]=>[1,3,4,2,6,5]=>1
[[[.,[.,.]],[.,.]],[.,.]]=>[2,1,4,3,6,5]=>2
[[[[.,.],.],[.,.]],[.,.]]=>[1,2,4,3,6,5]=>1
[[[.,[.,[.,.]]],.],[.,.]]=>[3,2,1,4,6,5]=>3
[[[.,[[.,.],.]],.],[.,.]]=>[2,3,1,4,6,5]=>2
[[[[.,.],[.,.]],.],[.,.]]=>[1,3,2,4,6,5]=>1
[[[[.,[.,.]],.],.],[.,.]]=>[2,1,3,4,6,5]=>2
[[[[[.,.],.],.],.],[.,.]]=>[1,2,3,4,6,5]=>1
[[.,[.,[.,[.,[.,.]]]]],.]=>[5,4,3,2,1,6]=>5
[[.,[.,[.,[[.,.],.]]]],.]=>[4,5,3,2,1,6]=>4
[[.,[.,[[.,.],[.,.]]]],.]=>[3,5,4,2,1,6]=>3
[[.,[.,[[.,[.,.]],.]]],.]=>[4,3,5,2,1,6]=>4
[[.,[.,[[[.,.],.],.]]],.]=>[3,4,5,2,1,6]=>3
[[.,[[.,.],[.,[.,.]]]],.]=>[2,5,4,3,1,6]=>2
[[.,[[.,.],[[.,.],.]]],.]=>[2,4,5,3,1,6]=>2
[[.,[[.,[.,.]],[.,.]]],.]=>[3,2,5,4,1,6]=>3
[[.,[[[.,.],.],[.,.]]],.]=>[2,3,5,4,1,6]=>2
[[.,[[.,[.,[.,.]]],.]],.]=>[4,3,2,5,1,6]=>4
[[.,[[.,[[.,.],.]],.]],.]=>[3,4,2,5,1,6]=>3
[[.,[[[.,.],[.,.]],.]],.]=>[2,4,3,5,1,6]=>2
[[.,[[[.,[.,.]],.],.]],.]=>[3,2,4,5,1,6]=>3
[[.,[[[[.,.],.],.],.]],.]=>[2,3,4,5,1,6]=>2
[[[.,.],[.,[.,[.,.]]]],.]=>[1,5,4,3,2,6]=>1
[[[.,.],[.,[[.,.],.]]],.]=>[1,4,5,3,2,6]=>1
[[[.,.],[[.,.],[.,.]]],.]=>[1,3,5,4,2,6]=>1
[[[.,.],[[.,[.,.]],.]],.]=>[1,4,3,5,2,6]=>1
[[[.,.],[[[.,.],.],.]],.]=>[1,3,4,5,2,6]=>1
[[[.,[.,.]],[.,[.,.]]],.]=>[2,1,5,4,3,6]=>2
[[[.,[.,.]],[[.,.],.]],.]=>[2,1,4,5,3,6]=>2
[[[[.,.],.],[.,[.,.]]],.]=>[1,2,5,4,3,6]=>1
[[[[.,.],.],[[.,.],.]],.]=>[1,2,4,5,3,6]=>1
[[[.,[.,[.,.]]],[.,.]],.]=>[3,2,1,5,4,6]=>3
[[[.,[[.,.],.]],[.,.]],.]=>[2,3,1,5,4,6]=>2
[[[[.,.],[.,.]],[.,.]],.]=>[1,3,2,5,4,6]=>1
[[[[.,[.,.]],.],[.,.]],.]=>[2,1,3,5,4,6]=>2
[[[[[.,.],.],.],[.,.]],.]=>[1,2,3,5,4,6]=>1
[[[.,[.,[.,[.,.]]]],.],.]=>[4,3,2,1,5,6]=>4
[[[.,[.,[[.,.],.]]],.],.]=>[3,4,2,1,5,6]=>3
[[[.,[[.,.],[.,.]]],.],.]=>[2,4,3,1,5,6]=>2
[[[.,[[.,[.,.]],.]],.],.]=>[3,2,4,1,5,6]=>3
[[[.,[[[.,.],.],.]],.],.]=>[2,3,4,1,5,6]=>2
[[[[.,.],[.,[.,.]]],.],.]=>[1,4,3,2,5,6]=>1
[[[[.,.],[[.,.],.]],.],.]=>[1,3,4,2,5,6]=>1
[[[[.,[.,.]],[.,.]],.],.]=>[2,1,4,3,5,6]=>2
[[[[[.,.],.],[.,.]],.],.]=>[1,2,4,3,5,6]=>1
[[[[.,[.,[.,.]]],.],.],.]=>[3,2,1,4,5,6]=>3
[[[[.,[[.,.],.]],.],.],.]=>[2,3,1,4,5,6]=>2
[[[[[.,.],[.,.]],.],.],.]=>[1,3,2,4,5,6]=>1
[[[[[.,[.,.]],.],.],.],.]=>[2,1,3,4,5,6]=>2
[[[[[[.,.],.],.],.],.],.]=>[1,2,3,4,5,6]=>1
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]=>[7,6,5,4,3,2,1]=>7
[.,[[.,[[[[.,.],.],.],.]],.]]=>[3,4,5,6,2,7,1]=>3
[.,[[[.,[[.,[.,.]],.]],.],.]]=>[4,3,5,2,6,7,1]=>4
[.,[[[.,[[[.,.],.],.]],.],.]]=>[3,4,5,2,6,7,1]=>3
[.,[[[[.,[.,[.,.]]],.],.],.]]=>[4,3,2,5,6,7,1]=>4
[.,[[[[.,[[.,.],.]],.],.],.]]=>[3,4,2,5,6,7,1]=>3
[.,[[[[[.,[.,.]],.],.],.],.]]=>[3,2,4,5,6,7,1]=>3
[.,[[[[[[.,.],.],.],.],.],.]]=>[2,3,4,5,6,7,1]=>2
[[.,.],[.,[.,[.,[.,[.,.]]]]]]=>[1,7,6,5,4,3,2]=>1
[[.,.],[.,[.,[.,[[.,.],.]]]]]=>[1,6,7,5,4,3,2]=>1
[[.,.],[.,[.,[[.,.],[.,.]]]]]=>[1,5,7,6,4,3,2]=>1
[[.,.],[.,[.,[[.,[.,.]],.]]]]=>[1,6,5,7,4,3,2]=>1
[[.,.],[.,[.,[[[.,.],.],.]]]]=>[1,5,6,7,4,3,2]=>1
[[.,.],[.,[[.,.],[.,[.,.]]]]]=>[1,4,7,6,5,3,2]=>1
[[.,.],[.,[[.,.],[[.,.],.]]]]=>[1,4,6,7,5,3,2]=>1
[[.,.],[.,[[.,[.,.]],[.,.]]]]=>[1,5,4,7,6,3,2]=>1
[[.,.],[.,[[.,[.,[.,.]]],.]]]=>[1,6,5,4,7,3,2]=>1
[[.,.],[.,[[.,[[.,.],.]],.]]]=>[1,5,6,4,7,3,2]=>1
[[.,.],[[.,.],[.,[.,[.,.]]]]]=>[1,3,7,6,5,4,2]=>1
[[.,.],[[.,[.,[.,[.,.]]]],.]]=>[1,6,5,4,3,7,2]=>1
[[.,[.,.]],[.,[.,[.,[.,.]]]]]=>[2,1,7,6,5,4,3]=>2
[[.,[.,[.,[.,[.,.]]]]],[.,.]]=>[5,4,3,2,1,7,6]=>5
[[.,[[[[.,.],.],.],.]],[.,.]]=>[2,3,4,5,1,7,6]=>2
[[[.,[[[.,.],.],.]],.],[.,.]]=>[2,3,4,1,5,7,6]=>2
[[[[[[.,.],.],.],.],.],[.,.]]=>[1,2,3,4,5,7,6]=>1
[[.,[.,[.,[.,[.,[.,.]]]]]],.]=>[6,5,4,3,2,1,7]=>6
[[.,[.,[.,[.,[[.,.],.]]]]],.]=>[5,6,4,3,2,1,7]=>5
[[.,[.,[.,[[.,.],[.,.]]]]],.]=>[4,6,5,3,2,1,7]=>4
[[.,[.,[.,[[.,[.,.]],.]]]],.]=>[5,4,6,3,2,1,7]=>5
[[.,[.,[.,[[[.,.],.],.]]]],.]=>[4,5,6,3,2,1,7]=>4
[[.,[.,[[.,.],[.,[.,.]]]]],.]=>[3,6,5,4,2,1,7]=>3
[[.,[.,[[.,.],[[.,.],.]]]],.]=>[3,5,6,4,2,1,7]=>3
[[.,[.,[[.,[.,.]],[.,.]]]],.]=>[4,3,6,5,2,1,7]=>4
[[.,[.,[[.,[.,[.,.]]],.]]],.]=>[5,4,3,6,2,1,7]=>5
[[.,[.,[[.,[[.,.],.]],.]]],.]=>[4,5,3,6,2,1,7]=>4
[[.,[[.,.],[.,[.,[.,.]]]]],.]=>[2,6,5,4,3,1,7]=>2
[[.,[[.,[.,[.,[.,.]]]],.]],.]=>[5,4,3,2,6,1,7]=>5
[[.,[[[[[.,.],.],.],.],.]],.]=>[2,3,4,5,6,1,7]=>2
[[[[[[.,.],.],.],.],[.,.]],.]=>[1,2,3,4,6,5,7]=>1
[[[.,[.,[.,[.,[.,.]]]]],.],.]=>[5,4,3,2,1,6,7]=>5
[[[.,[[[[.,.],.],.],.]],.],.]=>[2,3,4,5,1,6,7]=>2
[[[[.,[.,[.,[.,.]]]],.],.],.]=>[4,3,2,1,5,6,7]=>4
[[[[.,[[[.,.],.],.]],.],.],.]=>[2,3,4,1,5,6,7]=>2
[[[[[.,[.,[.,.]]],.],.],.],.]=>[3,2,1,4,5,6,7]=>3
[[[[[.,[[.,.],.]],.],.],.],.]=>[2,3,1,4,5,6,7]=>2
[[[[[[.,.],[.,.]],.],.],.],.]=>[1,3,2,4,5,6,7]=>1
[[[[[[.,[.,.]],.],.],.],.],.]=>[2,1,3,4,5,6,7]=>2
[[[[[[[.,.],.],.],.],.],.],.]=>[1,2,3,4,5,6,7]=>1
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]=>[8,7,6,5,4,3,2,1]=>8
[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]=>[7,6,8,5,4,3,2,1]=>7
[.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]=>[6,5,8,7,4,3,2,1]=>6
[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]=>[7,6,5,4,8,3,2,1]=>7
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]=>[6,5,7,4,8,3,2,1]=>6
[.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]=>[3,5,8,7,6,4,2,1]=>3
[.,[.,[[[.,[.,.]],[.,.]],[.,.]]]]=>[4,3,6,5,8,7,2,1]=>4
[.,[[.,.],[.,[.,[[.,.],[.,.]]]]]]=>[2,6,8,7,5,4,3,1]=>2
[.,[[[.,.],.],[[[.,.],.],[.,.]]]]=>[2,3,5,6,8,7,4,1]=>2
[.,[[[.,[.,.]],[.,[.,.]]],[.,.]]]=>[3,2,6,5,4,8,7,1]=>3
[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]=>[7,6,5,4,3,2,8,1]=>7
[.,[[.,[.,[.,[[.,[.,.]],.]]]],.]]=>[6,5,7,4,3,2,8,1]=>6
[.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]=>[6,5,4,3,7,2,8,1]=>6
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]=>[5,4,6,3,7,2,8,1]=>5
[.,[[.,[[[.,[.,.]],[.,.]],.]],.]]=>[4,3,6,5,7,2,8,1]=>4
[.,[[[.,[.,.]],[[.,[.,.]],.]],.]]=>[3,2,6,5,7,4,8,1]=>3
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]=>[4,3,5,2,7,6,8,1]=>4
[.,[[[[.,[.,.]],[.,.]],[.,.]],.]]=>[3,2,5,4,7,6,8,1]=>3
[.,[[[[.,[[[.,.],.],.]],.],.],.]]=>[3,4,5,2,6,7,8,1]=>3
[.,[[[[[.,[.,[.,.]]],.],.],.],.]]=>[4,3,2,5,6,7,8,1]=>4
[.,[[[[[.,[[.,.],.]],.],.],.],.]]=>[3,4,2,5,6,7,8,1]=>3
[.,[[[[[[.,[.,.]],.],.],.],.],.]]=>[3,2,4,5,6,7,8,1]=>3
[.,[[[[[[[.,.],.],.],.],.],.],.]]=>[2,3,4,5,6,7,8,1]=>2
[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]=>[1,8,7,6,5,4,3,2]=>1
[[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]=>[1,7,8,6,5,4,3,2]=>1
[[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]=>[1,6,8,7,5,4,3,2]=>1
[[.,.],[.,[.,[.,[[.,[.,.]],.]]]]]=>[1,7,6,8,5,4,3,2]=>1
[[.,.],[.,[.,[.,[[[.,.],.],.]]]]]=>[1,6,7,8,5,4,3,2]=>1
[[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]=>[1,5,8,7,6,4,3,2]=>1
[[.,.],[.,[.,[[.,[.,[.,.]]],.]]]]=>[1,7,6,5,8,4,3,2]=>1
[[.,.],[[.,.],[[.,.],[[.,.],.]]]]=>[1,3,5,7,8,6,4,2]=>1
[[.,.],[[.,[.,.]],[.,[.,[.,.]]]]]=>[1,4,3,8,7,6,5,2]=>1
[[.,.],[[.,[.,[.,[.,.]]]],[.,.]]]=>[1,6,5,4,3,8,7,2]=>1
[[.,.],[[[.,[.,.]],[.,.]],[.,.]]]=>[1,4,3,6,5,8,7,2]=>1
[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]=>[2,1,8,7,6,5,4,3]=>2
[[.,[.,.]],[[.,[[.,[.,.]],.]],.]]=>[2,1,6,5,7,4,8,3]=>2
[[.,[.,.]],[[[.,[.,.]],[.,.]],.]]=>[2,1,5,4,7,6,8,3]=>2
[[[.,.],.],[.,[.,[[[.,.],.],.]]]]=>[1,2,6,7,8,5,4,3]=>1
[[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]=>[3,2,1,8,7,6,5,4]=>3
[[.,[.,[.,.]]],[[.,[.,.]],[.,.]]]=>[3,2,1,6,5,8,7,4]=>3
[[[[.,.],.],.],[[[[.,.],.],.],.]]=>[1,2,3,5,6,7,8,4]=>1
[[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]=>[4,3,2,1,8,7,6,5]=>4
[[.,[.,[[.,.],.]]],[.,[[.,.],.]]]=>[3,4,2,1,7,8,6,5]=>3
[[.,[[.,.],[.,.]]],[[.,.],[.,.]]]=>[2,4,3,1,6,8,7,5]=>2
[[.,[[.,.],[.,.]]],[[.,[.,.]],.]]=>[2,4,3,1,7,6,8,5]=>2
[[.,[[.,[.,.]],.]],[[.,.],[.,.]]]=>[3,2,4,1,6,8,7,5]=>3
[[.,[[.,[.,.]],.]],[[.,[.,.]],.]]=>[3,2,4,1,7,6,8,5]=>3
[[.,[[[.,.],.],.]],[[[.,.],.],.]]=>[2,3,4,1,6,7,8,5]=>2
[[[.,[.,.]],[.,.]],[.,[.,[.,.]]]]=>[2,1,4,3,8,7,6,5]=>2
[[[.,[.,.]],[.,.]],[[.,.],[.,.]]]=>[2,1,4,3,6,8,7,5]=>2
[[[.,[.,.]],[.,.]],[[.,[.,.]],.]]=>[2,1,4,3,7,6,8,5]=>2
[[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]=>[5,4,3,2,1,8,7,6]=>5
[[.,[[.,[.,.]],[.,.]]],[.,[.,.]]]=>[3,2,5,4,1,8,7,6]=>3
[[[[.,[.,[.,.]]],.],.],[.,[.,.]]]=>[3,2,1,4,5,8,7,6]=>3
[[[[.,[[.,.],.]],.],.],[[.,.],.]]=>[2,3,1,4,5,7,8,6]=>2
[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]=>[6,5,4,3,2,1,8,7]=>6
[[.,[[.,[[.,[.,.]],.]],.]],[.,.]]=>[4,3,5,2,6,1,8,7]=>4
[[.,[[[.,[.,.]],[.,.]],.]],[.,.]]=>[3,2,5,4,6,1,8,7]=>3
[[.,[[[[[.,.],.],.],.],.]],[.,.]]=>[2,3,4,5,6,1,8,7]=>2
[[[.,[.,.]],[.,[.,[.,.]]]],[.,.]]=>[2,1,6,5,4,3,8,7]=>2
[[[.,[.,.]],[[.,.],[.,.]]],[.,.]]=>[2,1,4,6,5,3,8,7]=>2
[[[.,[.,.]],[[.,[.,.]],.]],[.,.]]=>[2,1,5,4,6,3,8,7]=>2
[[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]=>[4,3,2,1,6,5,8,7]=>4
[[[.,[[.,.],[.,.]]],[.,.]],[.,.]]=>[2,4,3,1,6,5,8,7]=>2
[[[.,[[.,[.,.]],.]],[.,.]],[.,.]]=>[3,2,4,1,6,5,8,7]=>3
[[[[.,[.,.]],[.,.]],[.,.]],[.,.]]=>[2,1,4,3,6,5,8,7]=>2
[[[[[[.,[.,.]],.],.],.],.],[.,.]]=>[2,1,3,4,5,6,8,7]=>2
[[[[[[[.,.],.],.],.],.],.],[.,.]]=>[1,2,3,4,5,6,8,7]=>1
[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]=>[7,6,5,4,3,2,1,8]=>7
[[.,[.,[.,[.,[.,[[.,.],.]]]]]],.]=>[6,7,5,4,3,2,1,8]=>6
[[.,[.,[.,[.,[[.,.],[.,.]]]]]],.]=>[5,7,6,4,3,2,1,8]=>5
[[.,[.,[.,[.,[[.,[.,.]],.]]]]],.]=>[6,5,7,4,3,2,1,8]=>6
[[.,[.,[.,[.,[[[.,.],.],.]]]]],.]=>[5,6,7,4,3,2,1,8]=>5
[[.,[.,[.,[[.,.],[.,[.,.]]]]]],.]=>[4,7,6,5,3,2,1,8]=>4
[[.,[.,[.,[[.,[.,[.,.]]],.]]]],.]=>[6,5,4,7,3,2,1,8]=>6
[[.,[[.,[.,.]],[.,[.,[.,.]]]]],.]=>[3,2,7,6,5,4,1,8]=>3
[[.,[[.,[.,[.,[.,.]]]],[.,.]]],.]=>[5,4,3,2,7,6,1,8]=>5
[[.,[[[.,[.,.]],[.,.]],[.,.]]],.]=>[3,2,5,4,7,6,1,8]=>3
[[.,[[[[[[.,.],.],.],.],.],.]],.]=>[2,3,4,5,6,7,1,8]=>2
[[[[.,.],[.,[.,.]]],[.,[.,.]]],.]=>[1,4,3,2,7,6,5,8]=>1
[[[[.,.],[[.,.],.]],[[.,.],.]],.]=>[1,3,4,2,6,7,5,8]=>1
[[[[[.,.],[.,.]],[.,.]],[.,.]],.]=>[1,3,2,5,4,7,6,8]=>1
[[[[[[.,.],.],.],[.,.]],[.,.]],.]=>[1,2,3,5,4,7,6,8]=>1
[[[[[[.,.],[.,.]],.],.],[.,.]],.]=>[1,3,2,4,5,7,6,8]=>1
[[[[[[[.,.],.],.],.],.],[.,.]],.]=>[1,2,3,4,5,7,6,8]=>1
[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]=>[6,5,4,3,2,1,7,8]=>6
[[[.,[[[[[.,.],.],.],.],.]],.],.]=>[2,3,4,5,6,1,7,8]=>2
[[[[[[.,.],.],[.,.]],[.,.]],.],.]=>[1,2,4,3,6,5,7,8]=>1
[[[[.,[.,[.,[.,[.,.]]]]],.],.],.]=>[5,4,3,2,1,6,7,8]=>5
[[[[.,[[[[.,.],.],.],.]],.],.],.]=>[2,3,4,5,1,6,7,8]=>2
[[[[[[.,.],[.,.]],[.,.]],.],.],.]=>[1,3,2,5,4,6,7,8]=>1
[[[[[[[.,.],.],.],[.,.]],.],.],.]=>[1,2,3,5,4,6,7,8]=>1
[[[[[.,[.,[.,[.,.]]]],.],.],.],.]=>[4,3,2,1,5,6,7,8]=>4
[[[[[[.,[.,[.,.]]],.],.],.],.],.]=>[3,2,1,4,5,6,7,8]=>3
[[[[[[[.,.],[.,.]],.],.],.],.],.]=>[1,3,2,4,5,6,7,8]=>1
[[[[[[[.,[.,.]],.],.],.],.],.],.]=>[2,1,3,4,5,6,7,8]=>2
[[[[[[[[.,.],.],.],.],.],.],.],.]=>[1,2,3,4,5,6,7,8]=>1
[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]=>[9,8,7,6,5,4,3,2,1]=>9
[.,[[[[[[[[.,.],.],.],.],.],.],.],.]]=>[2,3,4,5,6,7,8,9,1]=>2
[[[[[[[[[.,.],.],.],.],.],.],.],.],.]=>[1,2,3,4,5,6,7,8,9]=>1
[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]=>[10,9,8,7,6,5,4,3,2,1]=>10
[.,[[[[[[[[[.,.],.],.],.],.],.],.],.],.]]=>[2,3,4,5,6,7,8,9,10,1]=>2
[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]=>[1,2,3,4,5,6,7,8,9,10]=>1
[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]=>[8,7,6,5,4,3,2,1,9]=>8
[[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]],.]=>[9,8,7,6,5,4,3,2,1,10]=>9
[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],[.,.]]=>[2,1,4,3,6,5,8,7,10,9]=>2
[[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],[.,.]],[.,.]]=>[2,1,4,3,6,5,8,7,10,9,12,11]=>2
[[[[[[[[.,.],.],.],.],.],.],.],[.,.]]=>[1,2,3,4,5,6,7,9,8]=>1
[[[[[[[[[.,.],.],.],.],.],.],.],.],[.,.]]=>[1,2,3,4,5,6,7,8,10,9]=>1
[.,[[[[[[[.,[.,.]],.],.],.],.],.],.]]=>[3,2,4,5,6,7,8,9,1]=>3
[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],[.,.]]=>[1,2,3,4,5,6,7,8,9,11,10]=>1
[.,[[[[[[.,[[.,.],.]],.],.],.],.],.]]=>[3,4,2,5,6,7,8,9,1]=>3
[.,[[[[[[[[.,[.,.]],.],.],.],.],.],.],.]]=>[3,2,4,5,6,7,8,9,10,1]=>3
[.,[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]]=>[2,3,4,5,6,7,8,9,10,11,1]=>2
[[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]=>[1,9,8,7,6,5,4,3,2]=>1
[[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]],.]=>[7,8,6,5,4,3,2,1,9]=>7
[[.,.],[.,[.,[.,[.,[.,[[.,.],.]]]]]]]=>[1,8,9,7,6,5,4,3,2]=>1
[[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]=>[1,10,9,8,7,6,5,4,3,2]=>1
[[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]],.]=>[10,9,8,7,6,5,4,3,2,1,11]=>10
[[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]],.]=>[8,9,7,6,5,4,3,2,1,10]=>8
[[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]],.]=>[7,6,8,5,4,3,2,1,9]=>7
[[.,.],[.,[.,[.,[.,[[.,[.,.]],.]]]]]]=>[1,8,7,9,6,5,4,3,2]=>1
[[.,.],[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]=>[1,9,10,8,7,6,5,4,3,2]=>1
[[.,.],[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]=>[1,11,10,9,8,7,6,5,4,3,2]=>1
[[.,[.,.]],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]=>[2,1,10,9,8,7,6,5,4,3]=>2
[[.,[.,[.,[.,.]]]],[.,[.,[.,[.,[.,.]]]]]]=>[4,3,2,1,10,9,8,7,6,5]=>4
[[.,[.,[.,[.,[.,[.,.]]]]]],[.,[.,[.,.]]]]=>[6,5,4,3,2,1,10,9,8,7]=>6
[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],[.,.]]=>[8,7,6,5,4,3,2,1,10,9]=>8
[[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.],.]=>[7,6,5,4,3,2,1,8,9]=>7
[[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.],.]=>[6,5,4,3,2,1,7,8,9]=>6
[[[[[.,[.,[.,[.,[.,.]]]]],.],.],.],.]=>[5,4,3,2,1,6,7,8,9]=>5
[[[[[[.,[.,[.,[.,.]]]],.],.],.],.],.]=>[4,3,2,1,5,6,7,8,9]=>4
[[[[[[[.,[.,[.,.]]],.],.],.],.],.],.]=>[3,2,1,4,5,6,7,8,9]=>3
[[[[[[[[.,[.,.]],.],.],.],.],.],.],.]=>[2,1,3,4,5,6,7,8,9]=>2
[[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.],.]=>[8,7,6,5,4,3,2,1,9,10]=>8
[[[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.],.],.]=>[7,6,5,4,3,2,1,8,9,10]=>7
[[[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.],.],.]=>[6,5,4,3,2,1,7,8,9,10]=>6
[[[[[[.,[.,[.,[.,[.,.]]]]],.],.],.],.],.]=>[5,4,3,2,1,6,7,8,9,10]=>5
[[[[[[[.,[.,[.,[.,.]]]],.],.],.],.],.],.]=>[4,3,2,1,5,6,7,8,9,10]=>4
[[[[[[[[.,[.,[.,.]]],.],.],.],.],.],.],.]=>[3,2,1,4,5,6,7,8,9,10]=>3
[[[[[[[[[.,[.,.]],.],.],.],.],.],.],.],.]=>[2,1,3,4,5,6,7,8,9,10]=>2
[[[[[[[[[[.,[.,.]],.],.],.],.],.],.],.],.],.]=>[2,1,3,4,5,6,7,8,9,10,11]=>2
[[.,[[[[[[[.,.],.],.],.],.],.],.]],.]=>[2,3,4,5,6,7,8,1,9]=>2
[[[.,[[[[[[.,.],.],.],.],.],.]],.],.]=>[2,3,4,5,6,7,1,8,9]=>2
[[.,[[[[[[[[.,.],.],.],.],.],.],.],.]],.]=>[2,3,4,5,6,7,8,9,1,10]=>2
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Description
The first entry of the permutation.
This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1].
This statistic is related to the number of deficiencies St000703The number of deficiencies of a permutation. as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals
$$ \frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1). $$
This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1].
This statistic is related to the number of deficiencies St000703The number of deficiencies of a permutation. as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals
$$ \frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1). $$
Map
to 312-avoiding permutation
Description
Return a 312-avoiding permutation corresponding to a binary tree.
The linear extensions of a binary tree form an interval of the weak order called the Sylvester class of the tree. This permutation is the minimal element of this Sylvester class.
The linear extensions of a binary tree form an interval of the weak order called the Sylvester class of the tree. This permutation is the minimal element of this Sylvester class.
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